Joint Estimation of State and Sensor Systematic Error in Hybrid System

Transcription

1 Joint Estiation of State and Sensor Systeatic Error in Hybrid Syste Lin Zhou, Quan Pan, Yan Liang, Zhen-lu Jin School of utoation Northwestern Polytechnical University Xi an, 77, China bstract Consider the hybrid systes with nonlinear property, and the sensor easureents with unnown and tie-varying systeatic error in this paper. In order to obtain the joint least square (LS) estiation of state and systeatic error, a new ethod JE-EM (joint estiation-expectation axiization) is proposed. In this paper, the relationship between the sensor systeatic error estiation and state estiation is derived, which can be described by the fraewor of EM. Due to the character of the hybrid syste, the target state is estiated by the IMM with PF filter. ased on the above relationship, systeatic error is iteratively estiated by the fraewor of EM. Siulation results with a aneuvering target tracing scenario show the effectiveness of the proposed ethod. Keywords- Hybrid syste;systeatic error;interacting ultiple odel (IMM);Particle filter (PF);Expectation axiization (EM) I. INTRODUCTION In the integrated air surveillance syste, data fusion is an iportant issue in the sensor networ research field. The sensor easureents are collected by distributed sensors. However, sensor easureents include rando and systeatic error, and if aforeentioned easureents are not odified, these errors lead to degradation in trac inaccuracy perforance and even ay bring out ghost targets. In order to iprove the fusion precise of ultiple sensors, it is vital to eliinate errors. Rando errors can be solved by soe filtering algoriths, and sensor alignent ethods are used to deal with systeatic error. Sensor alignent is referred to as the registration proble, and it is an inherent proble in ulti-sensor syste, besides, it deals with the correction of registration errors. The sensor easureents after registered are vital that they are concerned with data correlation, trac anageent, target detection and filtering, classify and tracing, and so on. Many successful sensor registration algoriths have been proposed in literatures. The standard registration approach is the real tie quality control (RTQC) []. In [], Leung developed the least square (LS) ethod. The generalized least square (GLS) is given by ar-shalo [3], and Zhou advanced an algorith which was GLS based on earth-centered earth-fixed (ECEF-GLS) [4, 5]. esides, in [6], the exact axiu lielihood (EML) ethod incorporated the effects of easureent noise was proposed in 3. The aforeentioned algoriths are offline registration ethods. t the sae tie, there are soe online registration ethods. The Kalan filter (KF) [6] was usually used to estiate systeatic error. The extended Kalan filter (EKF) [8, 9] and the particle filter (PF) [5], which usually used to estiate the states of the augented in nonlinear syste. The siulated annealing and particle swar optiization (S-PSO) was proposed in [], it solved the proble of variant systeatic error registration, and overcae the defect of local optiu estiation, besides, avoided population degradation on particles. In [], Ignagni assued target state and systeatic error are initially uncorrelated, and developed the decoupled Kalan filter (DKF), which perfored two estiation schees including target state and systeatic error. In [, 3], the exact ethod (EX) and iproved EX were provided, the systeatic error estiation were accoplished by the local state estiation such that they yielded pseudoeasureents of the systeatic error with additive noises. In [4], He expressed the effect caused by radar systeatic error as rotation and ass otion between target tracs fro radars, and he proposed the Fourier transforation approach to estiate and copensate the rotation and ass otion between the target tracs and fusion center. esides, a series of ethods have been provided to jointly estiate target state and systeatic error. In [6], the augented state Kalan filter (SKF) ethod stac target state and sensors systeatic error into a single vector. In [7], Li had given the augented state unscented Kalan filter (SUKF), which was used to estiate the state of the augented nonlinear syste. In ultisensory surveillance, the exact axiu Kalan filter (EM-KF) [8] approach was provided to jointly solve data association, registration, and data fusion. In [9], Oello gave the axiu lielihood registration (MLR) with the arbitrary target state otion, the arbitrary nuber and types of sensors. These ethods have been used to solve sensor registration. However they are applied in the very restricted conditions. For instance, tie invariant systeatic error, siplified otion odel of target, nown statistic distribution of systeatic noise, and so on. In practice, any factors which affect sensors systeatic error are uncertainty. For exaple, the cliate, the terrain, the ray of different region, the property of nonlinear and ultiple odels, and so on. So, aforeentioned ethods are no longer adequate to solve sensor registration, and even worse can cause the fresh easureent errors. In the real applications, the transforation between otion odels has the property of stochastic or eergence which This paper is supported by National Natural Science Foundation of China (No. 635 No No. 6759). 969

2 leads to the syste with the character of nonlinear and ultiple odels. The aforeentioned syste is called hybrid syste, i.e. the traditional algoriths are not adequate to estiate systeatic error in hybrid syste. Otherwise, systeatic error estiation will produce serious deviation and decrease the precision of target tracing. The hybrid syste has been recently applied in ilitary civil fields with coplex environent. In order to iprove the precision of capturing and odify easureents, any aneuvering target tracing techniques have been developed. Either the target state or the systeatic error can be estiated in these ethods. Nevertheless, ost of traditional ethods cannot deal with the interaction of target state and systeatic error, for exaple, SKF or SUKF, and so on. To solve the proble, Song and Li [6,7] give the joint estiation ethods of state and systeatic error by EKF and UKF algoriths, however, syste noise is liited to a certain case. Li [8] proposed an algorith with the joint of data association, registration, and fusion using EM-KF, however, the ultiple odels of syste was not taen into account. Consider the hybrid syste with nonlinear property and ultiple odels, a new joint estiation-expectation axiization ethod is proposed. The target state and sensor systeatic error are jointly estiated here. In order to iprove the estiation results and solve the proble of nonlinear and the ultiple odels, both the EM, and IMM with PF are applied in this wor.this paper is organized as follows. In section II, we derive the relationship between state estiation and systeatic error estiation. We describe the steps of algorith in section III. Section IV illustrates the perforance of feasibility and efficiency based on the siulation scene. II. PRMETER ESTIMTION. Proble Stateent It is assued that target state contains ultiple odels, and sensor easureents involve systeatic error and nonlinear. In a hybrid stochastic syste with additive noise can be expressed by x = F x + Γ u =,,, M. (),,, where x represents target state at tie, and a noisy sensor easureent with additive systeatic error has the for z = h ( x) + b + v. () where represents the target otion odel and it is a hoogeneous Marov chain with transition probability as follows P{ }= π. (3) where,,,, M, and M is the set of odal states. F, is the nown state transition atrix, Γ, is the input control atrix related to odel. u, is the process noise with zero-ean and Gaussian covariance. The nonlinear vector function atrix h is a nown Q, s ri atrix ( r i represents state diension with odel). The sensor systeatic error b is a s vector, and it is independent of u,. The noise v is a Gaussian rando vector with zeroean and covariance R, and it is independent of u.,. Relationship of State Estiation and Systeatic Error Estiation In assued hybrid syste, target state and systeatic error are unnown paraeters. Due to the unnown systeatic error estiation, it is ipossible to estiate state by classic filter ethod i.e. KF, SKF, and so on. In order to jointly estiate state and systeatic error, it is necessary to give the relationship of state estiate and systeatic error estiation. We rewrite () as follows b = z h x + v = z h xˆ + h x h xˆ + v. (4) The systeatic error estiation can be calculated by the conditional expectation of systeatic error ˆ K b = E b Z K = z h xˆ + E h x Z h x ˆ. (5) ( ) if the condition blow is satisfied Eh ( K x Z ) = E h ( x ˆ). (6) then equation (5) can be rewrote as = z h xˆ. (7) is the least square (LS) estiation at the oent. C. EM pproach to Recursive Estiation The state estiation xˆ can be estiated by the filter. f xˆ, z, b ˆ as filter, and substitute it into (7), we Denote ( ) have ( ( ˆ ˆ,, )) = z h f x z b. (8) where xˆ is state estiation at tie. The equation (8) is nonlinear, so we copute systeatic error estiation by iterative ethod as follows, and ipleent until it converges to perfect results ˆ ( p+ ) ˆ,, ˆ ( p b = z h f x z b ). (9) ( ( )) where p corresponds to the p -th iteration. Due to the state estiation xˆ can be estiated by the filter ( ˆ ˆ,, ) the relationship between iteration has the for xˆ and ( ˆ ˆ,, ) f x z b, f x z b at the p -th ( ˆ p ) xˆ = f xˆ, z, b. () ( p ) Due to the expectation axiization (EM) [] algorith is an iterative procedure that estiates both the paraeters 97

3 and the issing or unobservable data during an iteration. Now, the joint estiation can be described by the two iterative steps E-Step: M-Step: ( ˆ p { )} xˆ = E f xˆ, z, b. () ( p ) ( ˆ p { ( ˆ ))} = arg ax z h f x, z, b. () ( p+ ) ˆ( p ) In the start of iteration, we replace b with initialization estiation b initial, and calculate state estiation x ˆ in the E- f xˆ, z, b ˆ, and update systeatic error Step by the filter ( ) estiation in the M-Step. D. State Estiation by IMM-PF Due to the hybrid syste with ultiple otion odels, in order to estiate state, the IMM tracer is the best solution to solve proble. It is necessary to interactively process the state estiation according to the odel transition probability π fro - to. Then, we copute the odel ixing probability π and the state ixing estiation x, with odel at tie. It is assued that the syste has nonlinear property and contains non-gauss syste noise, and then this paper adopts the particle filter (PF). The PF avoids soe probles including linearization error, syste noise restriction, and so no. In fact, the PF is the ethod of the optiization recursion ayesian filter based on Monte Carlo siulation. fter generating x,, we consider x, as the input () () of PF, and obtain particles x,,,,, x x,. It is necessary to update the above particles and obtain the () () ( N ) prediction x,,,,, x x, at tie. Note that N is the particle nuber with odel. ( N ) For odel, we have the output of PF x ˆ as N ( n) =, n= x xˆ x N. (3) fter getting ˆ, we utilize the odel probability odel and copute the fusion state estiation u with M = u ˆ = xˆ x. (4) where u is the updated odel probability, and it is coputed by the equation as follows u = Λ c. (5) c M where c = π u and c = Λ = M = lielihood probability Λ is as follows c T -. The odel Λ = exp{ v S v }. (6) s ( π) S S N ( n) (, ) ˆ = N n= N ( () ) ( () T n ˆ n,, ) ˆ = N n= In (6), the easureent residual v and the estiation covariance can be coputed by the denoted expression v z h x b. (7) S z h x b z h x b. (8) Note that is the unnown systeatic error estiation in (7) and (8), therefore, v and S cannot be directly coputed. However, we can replace with arbitrary initialization b initial. Substituting the arbitrary initialization of systeatic error in E-Step, we get the state estiation xˆ by the IMM-PF ethod. III. JOINT ESTIMTION LGORITHM OF STTE ND SYSTEMTIC ERROR In order to jointly estiate the state and the systeatic error, the target state estiation can be coputed by using particle filter (PF) based on interacting ultiple odel (IMM) at first. Then, the relationship between the sensor systeatic error estiation and state estiation is derived by above EM steps, and it is used for the optiizing process of joint estiation. The steps of the proposed algorith as follows: Paraeters initialization; FOR =,3,, K Model conditional re-initialization <> Copute the odel ixing probabilityπ using u and π ; <> Copute the state ixing estiation x, using () the interaction input state x ˆ, the state () P, covariance, and π,; Setting repetition label flag ; WHILE e Maxn, and flag is true Setting repetition label flagcore ; WHILE e < 3 or flagcore is true Model conditional filtering <> State saple:, 97

4 Randoly saple fro x, to get the state particles,,,,,,, N,, x x x ; <> The state particle update: Transit particles x,,, x,,,, x N,, to ( e) ( e) ( e),,,,,,, N,, x x x ; Re-saple ( e) ( e) ( e),,,,,,, N,, x x x to obtain ( e ) ( e ) ( e ),,,,,,, N,, x x x ; Copute the state estiation x ˆ, the easureent residual covariance v S using (3), (7), (8);, and the estiation Copute the lielihood function of easureent by (6) Model probability update: Update the state odel probability u by (5); Estiation fusion Copute the fusion state estiation x using x ˆ, P, and u, and (4); ˆ( e ) ˆ e Calculating systeatic error b by (); Condition judgent: Let e = e+ ; if e 3 then reset flagcore ; END WHILE Condition judgent: Let e = e ; If ˆ ( e ) ˆ ( e b = b ) or e = Maxn then we calculate the fusion systeatic error estiation and consider it as the initialization b + at tie +, besides, reset flag ; Update repetition paraeters : Reset flagcore, and Let e = e+ ; END WHILE END FOR where e expresses the EM iterative tie, and Maxn is the axiu iterative tie. IV. SIMULTION RESULTS. Proble Stateent In this section, it is assued that the otion of the aneuver target is restricted to the horizontal plane. We desire to trac target based on the polar easureents fro the radar, and the easureents contain systeatic error. In hybrid syste with additive stochastic noise, the target state evolves according to the odel as follows Fx + Γu, 3 x = Fx + Γu, 3 < 4 Fx + Γu, 4 < Two different state transition atrixes are given as follows, and τ = is the sapling tie interval τ F = τ sin( ωτ ) ω (cos( ωτ ) ) ω cos( ωτ ) sin( ωτ ) F = ( cos( ωτ )) ω sin( ωτ ) ω sin( ωτ ) cos( ωτ ) There, Γ is input control atrix τ τ Γ = τ τ Let x [ ] T = x x y y be the state vector of target, and x, y, x, y represents the orthogonal coordinates and coordinates direction velocity of the horizontal plane, respectively. Note that between scans and 3, the target is constant velocity otion, and velocity coponent is x =.3 / s, y =. / s, respectively. Fro scan 3 to scan 4, the target is aneuvering curve otion, and angular velocity ω is.5rad/s. fter scan 4, the target is such otion as the first period. The process noise u, and u, are belonging to the above different odels, the covariances are given below Q = diag([.6,.6 ]), Q = Q In polar coordinate, the easureent odel with additive systeatic error has the for z h ( x ) Z = = + b + v z h ( x ) = = ( x x) + ( y y) r = = y y θ tan x x x y y is state vector with initialization ( x x) + ( y y) r ( ) y y θ tan x x h x h x where [ ] T x = x T x = [35.3 / s], T x = [. / s] ;( x, y) and ( x, y ) are the location of two radars 97

5 with x = 5, y =, x = 3, y = 65 ; T = [ Δ, Δ ] r θ b b b is systeatic error vector and Δ b = [ Δ ; Δ ], Δ b = [ Δr ; Δθ ]; v is a white, zeroean Gaussian sequence with nown covariance R = diag([ R ]), R = R θ = diag([(. ) (. ) ]) ; Z r R θ T = [ r, θ, r, θ ] r is the sensor easureent vector in polar plane, r, θ, r, θ are the ranges and aziuths of sensor and, respectively. The transition probability of Marov chain is π ij = [.99.;..99] in IMM, and the probability atrix of two odels is u = [.5.5]. The initialization target T state estiation is xˆ = x ˆ = [ x,, y,], where z z xz and Z are the orthogonal coordinates of easureent at first saple tie, and initialization covariance atrix is P = diag([ Pinit P init ]), P init = [.. / s ].The sapler particle nuber N is 3. The siulation step is 3 6. The EM iterative tie is. esides, the Monte Carlo tie is 5.. Siulation Results ccording to given paraeters, target real trajectory of single step is shown in Figure. y() Trac-Real Sensor Sensor x() Figure. Target real trajectory It is assued that the systeatic error is tie-varying, and it has expression b = + F b + b v b where Fb is systeatic error transition atrix, and it is T diag([ I I ]) ; systeatic error vector b = [ b, b ], it has initialization value [,,.5,.8 ] T ; systeatic error b noise v is a white, zero-ean Gaussian sequence with nown covariance R b = diag([(.5 ) (.8 ) (.8 ) (.6 ) ]). We get the corresponding results of tie-varying systeatic error based on assued paraeters shown in Figure. y z Range-error() Range-error().5.5 Sensor ziuth-error( ) ziuth-error( ).5.5 Sensor Figure. Systeatic error on range and aziuth fter jointly estiate state and systeatic error based on JE-EM proposed in this paper, it ipleents the systeatic error registration of sensor and. fter single step, the Figure 3 gives the coparison trajectory aong real, unregistered, and registered easureent. It shows that the registered easureent is better than the unregistered. y() Real Measureent- Measureent- Registrated easureent- Registrated easureent- Sensor Sensor x() Figure 3. The coparison trajectory of real, unregistered easureent, and registered easureent We obtain the systeatic error estiation using the proposed ethod. Figure 4 gives the coparison of estiation and assued systeatic error. Range-error() Range-error() Real() Estiation - 3 Real() Estiation ziuth-error( ) ziuth-error( ) 3 Real() Estiation - Real() Estiation - Figure 4. The systeatic error coparison of real and estiation 973

6 Range-error() Range-error() Sensor ziuth-error( ) ziuth-error( ) - Sensor Figure 5. Systeatic error deviation of real and estiation ccording to Figure 4, the Figure 5 is the systeatic error deviation of systeatic error estiation and assued systeatic error. In Figure 5, the range deviation scope is ~., the aziuth deviation approxiately is., and sensor and have average range deviations.45,.6, and aziuth deviation.,., respectively. Copare the range deviation and aziuth deviation with the assued systeatic errors resolution (sensor has resolution.5 and.8, sensor has resolution.8 and.6 ), the above deviations are uch lesser. Siulation results reveal that the JE-EM algorith can preferably estiate systeatic error of sensor and sensor. In conclusion, utilizing the iterative property of EM and the relationship between state estiation and systeatic error estiation, we can iteratively update state estiation and systeatic error estiation at every sapling tie. fter 5 ties Monte Carlo, Figure 6 gives the RMSE of distance, velocity on X and Y axes, respectively. In Figure 6, the RMSE of target state estiation have sharp decline excluding the velocity of Y axes on sensor at the start. The average runtie is 56.73s after 5 ties siulation, and the average RMSE of target state estiation on X and Y axes as TLE I. RMSE-X() RMSE-Y() Sensor Sensor- 5 5 RMSE-Vx(/s) RMSE-Vy(/s) Sensor Sensor- 5 5 (a) Sensor RMSE-X() RMSE-Y() TLE I. RMSE RMSE-Vx(/s) RMSE-Vy(/s) (b) Sensor Figure 6. RMSE of target state estiation X axes () RMSE VERGE OF TRGET STTE ESTIMTION X axes velocity (/s) Y axes () Y axes velocity (/s) Sensor Sensor The RMSE of tie-varying systeatic error after 5 ties Monte Carlo as Figure 7. In Figure 7, we can see the RMSE of systeatic errors have a sharp decline at the start. Subsequently, curve preservers around the certain value. RMSE-Range() RMSE-ziuth( ) RMSE-Range() RMSE-ziuth( ).5 Sensor Sensor (a) Sensor (b) Sensor Figure 7. RMSE of systeatic error 974

7 ccording the RMSE of sensor systeatic error estiation after 5 ties Monte Carlo, we get the average RMSE of range errors and aziuth errors as follows in TLE II. TLE II. RMSE VERGE OF SYSTEMTIC ERROR ESTIMTION RMSE Range error() ziuth error( ) Sensor.79.8 Sensor.8.3 V. CONCLUSION In the real applications, a lot of stochastic or eergent factors cause the otion odels transferring in any hybrid systes, which produces serious estiation deviation and decreases tracing precision. new ethod called JE-EM is introduced to solve this proble. In this wor, the ultiple odels and the nonlinear property are taen into account in dynaic hybrid syste. We estiate state by IMM with PF ethod in this paper. Then, the relationship between the target state estiation and systeatic error estiation are derived, and this relationship can be described by the fraewor of EM, and it can be used to optiize the joint estiation. The proposed ethod is used to update the unregistered easureents with systeatic error. The updated easureents can provide ore reliable inforation to soe wors, i.e. easureent associating, target tracing, and so on. [] X.D. Lin, Y. ar-shalo, T.Kirubarajan, Exact ultisensor dynaic bias estiation with local tracs, IEEE Transactions on erospace and Electronic Systes, vol.39, no.4, 4, pp [3] X.D. Lin, Y. ar-shalo, T.Kirubarajan, Multisensor-ultitarget bias estiaiton for general asynchronous sensors, IEEE Transactions on erospace and Electronic Systes, vol.4, no.3, 6, pp [4] Y. He, Q. Song, W. Xiong, trac registration-correlation algorith based on Fourier transfor, cta eronautica Et stronautica Sinica, vol.3, no.,, pp [5]. Zia, T. Kirubarajan, P. Reilly, Jaes, D. Yee, et al, n EM algorith for nonlinear state estiation with odel uncertainties, IEEE Transactions on Signal Processing, vol.56, no.3, 8, pp [6] Q. Song, Y. He, Y.L. Dong, n joint estiation algorith for state and systeatic errors, Journal of Projectiles, Rocets, Missiles and Guidance, vol.7, no.4, 7, pp [7] W. Li, H. Leung, Y. Zhou, Space-tie registration of radar and ESM using unscented Kalan filter, IEEE Transactions erospace and Electronic Systes, vol. 4, no.3, 4, pp ,. [8] Z. H. Li, S. Y. Chen, H. Leung, Joint data association, registration, and fusion using EM-KF, IEEE Transactions on erospace and Electronic Systes, vol.46, no.,, pp [9] N. Oello,.Ristic, Maxiu lielihood registration for ultiple dissiilar sensors, IEEE Transactions on aerospace and electronic systes, vol.39, no.3, 3, pp [] P. Depster, N. Laird, D.Rubin, Maxiu lielihood fro incoplete data via the EM algorith, J. Roy. Stat., vol.39, 977, pp REFERENCES [] J J. ure, The SGE real tie quality control function and its interface with UIC II/UIC-III, MITRE Corporation Technical Report, No.38, London, England, Noveber 996. [] H. Leung, M. lanchett, least square fusion of ultiple radar data, Proceeding of Radar 94, Paris, 994, pp [3] Y.ar-Shalo, Multitarget-ultisensor tracing: advanced applications, in Registration:a prerequisite for ultiple sensor tracing, M. P.Dana, Norwood, M: rtech House, 99. [4] Y.F. Zhou, L. Henry,.Martin, Sensor alignent with earth-centered earth-fixed (ECEF) coordinate syste, IEEE Transactions on erospace and Electronic Systes, vol. 35, no., 993,pp [5] I.T.Li, J. Georganas, Multi-target Multi-platfor sensor registration in geodetic coordinates, Proceedings of the Fifth International Conference on Inforation Fusion, vol., no.,, pp [6] N. Oello,. Ristic, Maxiu lielihood registration for ultiple dissiilar sensors, IEEE Transactions on erospace and Electronic Systes, vol.39, no.3, 3, pp [7] S. Dhar, pplication of a recursive ethod for registration error correction in tracing with ultiple sensors, Proceeding of erican Control Conference, San Francisco, C, 993, pp [8] E. J. Dela Cruz,. T. louani, T. R. Rice, W. D. lair, Sensor registration in ultisensor systes, Proceeding of SPIE Conference on Signal and Data Processing of Sall Targets, Orlando, FL, US, 99, pp [9].Friedland, Treatent of bias in recursive filtering, IEEE Transactions on utoatic Control, vol.4, no.4, 969, pp [] L. Zhou, Q. Pan, Y. Liang, pplication of iproved S-PSO in the syste error registration, Opto-Electronic Engineering, vol.37, no.9,, pp [] M.. Ignagni, n alternate derivation and extension of Friedland s two-stage alan estiator, IEEE Transactions on utoatic Control, vol.6, no.3, 98, pp